[42.19] Using the Correlation Exponent as a Measure of Chaos in Celestial Mechanics

M. C. Lewis (Trinity U.), H. F. Levison (SwRI)

The correlation exponent is a measure of chaos that was initially used to
study dissipative systems. It gives a numeric value describing how a set is
distributed through a given space. Its creators (Grassberger amp; Procaccia,
Physica D, 9, 189) argue that while it returns values that are similar to the
fractal dimension, it is superior because it is sensitive to the density of the
set. It also has the advantage that it can be used to measure sets produced by
non-dissipative systems or fat fractals. We are investigating how the
correlation exponent can be used as a measure of the chaos of orbits in our
solar system. To initiate this work, we must determine the proper ranges of
input parameters to numerical integrations of the orbits and select the best
coordinate system for making accurate, precise measurements. A comparison of
the return values in different coordinate systems with the Lyapunov exponents
of massless test particles between Jupiter and Saturn will be presented.
Research done to date implies that 1) a strong relationship exists between the
Lyapunov exponent and the correlation exponent and 2) the correlation
exponent is a robust measure of chaos in celestial systems. The correlation
exponent also has the advantage that it can be calculated from a data file
of particle positions after the integration is complete. We would like to
acknowledge the AAS for funding this research with an NSF REU grant.